Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Paloma Bengoechea ¹   ; Nikolay Moshchevitin ²

¹ Department of Mathematics ETH Zürich Ramistrasse 101 8092 Zürich, Switzerland
² Faculty of Mathematics and Mechanics Moscow State University Leninskie Gory 1 GZ MGU, 119991 Moscow, Russia

Acta Arithmetica, Tome 177 (2017) no. 4, pp. 301-314

Résumé

For any $j_1,\ldots,j_n \gt 0$ with $\sum_{i=1}^nj_i=1$ and any $\theta\in\mathbb R^n$, let ${\mathrm{Bad}_{\theta}(j_1,\ldots,j_n)}$ denote the set of points $\eta\in\mathbb R^n$ for which $\max_{1\leq i\leq n}(\|q\theta_i-\eta_i\|^{1/j_i}) \gt c/q$ for some positive constant $c=c(\eta)$ and all $q\in\mathbb N$. These sets are the ‘twisted’ inhomogeneous analogue of $\mathrm{Bad}(j_1,\ldots,j_n)$ in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e. provided that $j_i=1/n$, and in the weighted setting when $\theta$ is chosen from $\mathrm{Bad}(j_1,\ldots,j_n)$. We generalise these results by proving the full Hausdorff dimension in the weighted setting without any condition on $\theta$. Moreover, we prove $\dim(\mathrm{Bad}_{\theta}(j_1,\ldots,j_n)\cap\mathrm{Bad}(1,0,\ldots,0)\cap\cdots\cap\mathrm{Bad}(0,\ldots,0,1))=n$.

Détail

DOI : 10.4064/aa8234-11-2016

Keywords: ldots sum theta mathbb mathrm bad theta ldots denote set points eta mathbb which max leq leq theta i eta positive constant eta mathbb these sets twisted inhomogeneous analogue mathrm bad ldots theory simultaneous diophantine approximation has shown have full hausdorff dimension non weighted setting provided weighted setting theta chosen mathrm bad ldots generalise these results proving full hausdorff dimension weighted setting without condition theta moreover prove dim mathrm bad theta ldots cap mathrm bad ldots cap cdots cap mathrm bad ldots